ANSWERS
Mathematics 4 (Mathematical Analysis)
page 1
Chapter 4
Big Ideas and Little Tricks: ALGEBRA FOR COLLEGE MATHEMATICS
COURSES
b) u = 4q – 2 or u = sin (4q – 2)
c) u = 3x
2
AL-1.
a) u = x 2 + 2
AL-2.
a) u = y-5/2, x = 1, y = 4-2/5, u = 4
b) u = x 2 + 3x , x = - 3± 2 409 , y = 1, u = 10
AL-3.
a) u + u – 6 = 0
b) v2 + v – 6 = 0
d) M = - 3±2 29 . No value associated with v = -3.
AL-4.
x + y, x2 + 2xy + y2, x3 + 3x2y + 3xy2 + y3,
x4 + 4x3y + 6x2y2 + 4xy3 + y4
AL-5.
a) Decrease by 1 each time
b) Increase by 1 each time
c) Each time the sum is the same as the exponent of expansion.
AL-6.
a) 25
AL-8.
1 goes in “Row 0.”
AL-9.
Row 9
7
b) 3x+5
c) 2x+1
x±2
x +3
c) v = -3 or 2
d) x-2
e) x + 1
AL-10. x9 + 9x8y
AL-11. a) 20
b) 5
c) 35
d) 252
AL-12. a) 1
b) 0
yπ0
e) Impossible. 4
c) 4
d) 1
y
f) Impossible. 4 π -3
AL-13. a) x3 – 6x 2 + 9x + 3
c) W + 25
AL-14. x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6
AL-15. 1, 8, 28, 56, 70, 56, 28, 8, 1
AL-16. Multiply top and bottom by xy2.
Al-17.
xy3 +x 2 y2
a)
1+x3y3
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x 2 y6 +1
b)
1+xy 3
v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
AL-18. a)
x 2 +y 2
xy 2
b)
page 2
xy 2
x2 + y2
c)
x2 y 4
(x 2 +y 2 )2
AL-19. a) xy+1
2
b) 3a b 2
AL-20. a) x2 + 3x2y + 3xy2 + y2
c) x4 + 12x3w + 54x2w2 + 108xw3 + 81w4
y
AL-22.
c)
a +b
a 4 b2
(a 3 +b 2 )2
x15 + 15x14 y + 105 x13 y 2 + 455 x12 y 3
AL-24. a) x3 + 3x2 + 3x + 1
4
AL-25. a) 3+4x
4
b) x3 + 6x2 + 12x + 8
b)
2x ±x
y 2 +x 2 y 4
x 4 y 2 ±x2
c)
AL-26. a) x3 + 3x2y + 3xy2 + y3
c) x3 + 3x2(-4w) + 3x(-4w)2 + (-4w)3
AL-27. a) xy2
b) y = -4w
d) x3 – 12x2w + 48xw 2 – 64w3
b) x6y-1
c) x6y-2 + y-1
AL-28. a) 180˚
b) 225˚
c) -60˚
AL-29. a) 9
b) 6
c) 60 units3
AL-30. a) p6 , 5p6 , 7p6 , 13p6
()
1+xy
x y ±xy2
3 2
d) -57.2957...˚
b) same as part (a)
( )
( )
AL-31. a) 37
x- 1
b) x+3
x+5
c) x+4
AL-32. a) 1
b) -1
c) -1
PROBLEM SET A
1.
()
10
120 12
ª 0.117
2.
3.
( 0.8) 4 ª 0.410
4.
5.
6( 0.2 )2 ( 0.8)2 ª 0.154
6.
7.
153( 0.9 )16 ( 0.1)2 ª 0.284
8.
1
AL-35. a) 16
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5( 0.6 ) 4 ( 0.4 ) ª 0.259
()()
3
3
20 ( 16 ) ( 65 ) ª
3
4 43 14 ª 0.422
0.054
10 ( 0.3)2 ( 0.7)3 ª 0.309
b) Can get red-blue or blue-red combination.
v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 3
AL-36. Total probability equals 1.
c) q2
b) p2
d) 2pq
AL-37. a) sin2u – sin u + 0.24 = 0
c) v2 – v + 0.24 = 0
b) Let v = sin u
d) v = sin(3x – 5)
AL-38. a4 + 4a3bc + 6a2b2c2 + 4ab3c3 + b4c4
AL-39. b) 0.73 = 0.343
d) 3(0.7)2(0.3) = 0.441
c) 0.33 = 0.027
e) 3(0.7)(0.3)2 = 0.189
AL-40. a) 2 + 6 3
b) 13
AL-41. a) 207
d) 2a4 + 5a2
AL-42. x =
c) b = 1
b) 207
e) They are the same.
3+3 2
ª 0.176, y =
p 3+ 3
c) 2a4 + 5a2
3-p 3 2
ª -1.239
p 3+ 3
AL-43. a) any angle in the 4th quadrant
b) 7p6 or 11p6
c) any angle in the 3rd quadrant
d) Approximately 2.8 radians works
e) No—it doesn’t satisfy the Fundamental Pythagorean Identity.
AL-44. a) 6 – 2i
AL-45.
b) -2 + 18i
c) 23 + 2i
- 7 22
+ i
41 41
AL-46. a) (x ± y)(x + y)
2
b) (x±1)
(x+1)
c)
1
x (3x+y)
3
AL-47. One contains x = 3 and the other does not.
x+y
AL-48. a) x±y
AL-49.
b) x ± 4
2
c) (x+1)
x+2
4+2 3
AL-51. a) 7 + 3 5
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7
b) 11±3
2
c) 17 - 5 11
v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 4
AL-52. a) a + 2 ab + b
a±b
b)
AL-53. a) 00 indeterminate
b)
a- b
a - 2 ab + b
1
x+h + x
c) 1
2 x
1
3x + h + 3x
AL-54.
AL-55. a) 3 - 1
b)
AL-56. a) 8
b) -8
x± x ±2
x- 4
c) a3
d) a3
e) f(-a) = -f(a)
AL-58. a) p3 + 3p2q + 3pq2 +q3
b) 3pq2
c) q3
AL-59. a) 7 + 3i
b) 4 – 2i
c) 27 ± 9i
b) 3 - 7i
26
AL-60. a) 18
25 ± 25 i
AL-61. a)
p
p q - q
c) -8i d) 33 or 1
3
b) sin q cos q
2 2
AL-62. 13 + i
AL-63. a)
AL-64
y2 - x2
xy
b)
y2 - x2
x2 y2
-2
AL-65. a) increasing x > 2; decreasing x < 2
AL-70. Increasing on (-• , -1) and (1, • ), decreasing on (-1, 0) and (0, 1);
concave up on (0, • ), concave down on (-• , 0).
AL-71. a) decreasing (-• , 0) and (0, • ), concave up (0, • ), concave down (-• , 0)
b) increasing (-• , • ), concave up (-• , • )
c) decreasing (-• , • ), concave up (-• , • )
d) decreasing (-• , • ), concave up (-• , 0), concave down (0, • )
AL-72. x3 – 12x is such a function.
AL-73. y = 5x, y = sin x, ...
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v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 5
AL-74. Shiloh; the segments are above the graph.
AL-75. It is the same.
AL-76. a) x = -2 + 10i
c) x = 2 + 3i
AL-77.
b) x = 2 – i
d) x = 1.5 + 0.5i
b) 3 2 + 3
30+22 2
17
AL-78. a) -18
b) 18
e) f(-a) = -f(a)
c) - 2 + 18 ± 12 + 6
c) -a3 + 3a
d) a3 – 3a
b) x2 + 2yz + x2; y3 + 3y2z + 3y2 +z3
AL-79. a) x3 + 3x2u + 3xu2 + u3
c) x 3 + 3x 2 y + 3x 2 z + 6xyz + 3xy 2 + 3xz 2 + 3y 2 z + 3yz 2 + y 3 + z 3
AL-80. x4 + 4x2 + 6 + 4x-2 + x-4
AL-81. b) It is a minimum because the rectangles are under the curve.
c) 0.4 Â
4
( 0.4 j + 1)2 + 5
j=0
AL-82. a)
3x
2
x - y2
b)
x 2 +y 2
x2 - y2
AL-83. b = 225.062
AL-84. -13
AL-85. a) 4, 4, 9, 9, 2.962, 2.962
b) Changing sign of x doesn’t affect f(x).
c) They are equal
d) f(-a) = f(a)
e) symmetric about y-axis
AL-86. a) even powers
b) f(-x) = f(x)
c) They are symmetric about the y-axis.
d) y = cos x or y = | x | are good choices.
AL-87. a) 8, -8, 27, -27, 5.097, -5.097
b) Changing the sign of x changes the sign of f(x).
c) opposite signs
d) f(-a) = -f(a)
e) symmetric about the origin
AL-88. a) odd powers
b) f(-x) = -f(x)
c) They are symmetric about the origin.
d) y = sin x is a good choice.
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v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
AL-89. a) (-2, 5)
b) (3, -5)
page 6
c) unknown
AL-92. b) (x1/2)6 is not defined. Others are.
c) If x = -4, x1/2 = 2i and (x1/2)6 = -64
AL-93. C = 6 or 12
AL-94. a) 1
b) 12 = 2 3
AL-95. a) 1
e) 49
4
b) 4
2
f) b4
c) 16
d) 25
AL-96. y = (x + 1)2, y = (x – 2)2, y = (x + 4)2, y = (x – 5)2,
(
)2
(
y = x + 27 , y = x + 2b
)2
AL-98. y = (x + 3)2 – 10
AL-99. The pens should be 37.5 feet wide and 50 feet long.
AL-100. a) u = sin q
b) u = -1, 4. 0 is not in the domain.
AL-101. a) 16
b) 16
c) 32
AL-102. a) 2
b) 71/60
c) q = 3p2
1
d) 16
AL-103. b) sine is odd, cosine is even
AL-104. a) even
b) neither
c) odd
AL-105. a) 9
b) y = (x + 3)2 – 6
c) V(-3, -6)
AL-106. V(4, -15)
AL-107. a) 27
b) y = 3(x – 3)2 – 26, V = (3, -26)
AL-108. a) y = 2(x – 2)2 – 1
b) (2, -1)
AL-109. (x + 3)2 + (y – 2)2 = 64; center: (-3, 2); radius: 8
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v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 7
13500
2
e) h = 2250
2 , S = 2x +
x
AL-110. c) S = 2x2 + 6xh; 4500 = 2x2h
x
f) x = 15, h = 10, S = 1350
AL-111. The only change is that S = 4x2 + 6xh. After solving, x = 315 ª 11.906,
h = 10 ◊3 4 ª 15.874, S = 900(4 - 1/3 + 21/3 ) ª 1700.893
2
AL-112. center: (-4, 9), radius: 120 ª 10.954
AL-113. a4 – 8a3b + 24a2b2 – 32ab3 + 16b4
AL-114. No, because the vertex must lie on the y-axis if it is even.
AL-115. x = 2 - 1
AL-116. a) even
AL-117. a) 2
d) 2
b) neither
b) -1
e) 7
c) odd
c) 10
f) 14
AL-118. They’re the same because 60° = p3 radians.
AL-119. a) C = (5, -4), r = 6
AL-120. a) x2 + y2 = 100
b) C = (4, -3), r = 9, shaded on inside
b) (x – 7)2 + (y – 5)2 = 65
5
AL-122. 2x2 – 2x + 2, remainder -5 or 2x2 – 2x + 2 – 2x+1
AL-123. 2x2 – 4x + 13
AL-124. It is. The quotient is x2 – 6x + 9 with no remainder.
AL-125. a) f -1(x) = x3 – 2
c) f -1(x) is a reflection of f(x) over the line y = x.
- 2x or 2x
AL-126. g-1(x) = x±2
2- x
AL-127. f(x) = (x – 6)1/3
AL-128. It is not. Remainder is 140.
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v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 8
AL-129. f -1(x) = 1- xx or x-- x1
d) x ≥ 0
AL-130. b) It won’t pass the vertical line test.
e) f -1(x) = 1-2xx
AL-131. 8 – 36x + 54x2 – 27x3
AL-132. a) x3 + 6x2y + 12xy2 + 8y3
b) x3 – 6x2y + 12xy2 – 8y 3
c) Every other sign is different. (-1)k changes every other sign.
AL-133. x = ±i 13 , y = 53
AL-134. a) ª 2.25
b) 42.4
8
b
AL-135. b ª 0.709, a = 329.28
AL-136. d; p = 0.4 or ª 0.779. Teacher Solution: The probability of exactly three heads
5
is p3(1 – p)2 which we want to equal 144 . That is, we need to solve the
()
625
3
equation 10p3(1 – p)2 = 144
625 . Solving graphically gives the required solutions.
È x4 + 1 ˘
AL-137. Í
˙
2
Î x+x ˚
AL-140. b) 2
d) 70
AL-141. Pascal’s Triangle
AL-145. a)
2: 1
b) x =
2
2 ±1
c) x = 2( 2 + 1)
AL-147. Let x = a + bi and so x2 = a2 – b2 + 2abi = i. Thus 2ab = 1 and
1+ i
a2 – b2 = 0. Thus x = ±
. Students that get seriously involved in this
2
problem might want to be told eventually about the fact that
eiq = cos q + i sin q which is an easy way to take roots.
AL-148. It is a line.
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v. 3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 9
AL-149. a) 8 - 2
b)
4 = 2
2+ 8 1+ 2
x 4 y6 +y3
AL-150. a) 2 3 3
x +x y
b) x
AL-151. It would not be a function because it fails the Vertical Line Test.
AL-152. Error is here. Parentheses should contain x2 – 2x.
AL-153. a) a parabola with a vertex on the y-axis
c) a parabola with a vertex not on the y-axis
b) impossible
AL-154. odd
AL-156. C = 6 or 12
AL-157. a) 12
AL-158. a) 2p3 , 5p3
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b) 22
c) -1
b) 5p6 , 7p6
d) 0
c) p6 , 5p6 , 7p6 , 11p6
v. 3.1